Exact Joint Density-Current Probability Function for the Asymmetric Exclusion Process

We study the asymmetric simple exclusion process with open boundaries and derive the exact form of the joint probability function for the occupation number and the current through the system. We further consider the thermodynamic limit, showing that the resulting distribution is non-Gaussian and that the density fluctuations have a discontinuity at the continuous phase transition, while the current fluctuations are continuous. The derivations are performed by using the standard operator algebraic approach and by the introduction of new operators satisfying a modified version of the original algebra.

Figure 1

Phase diagram of the one dimensional exclusion process. The dashed line indicates the first order transition line, while the dash-dotted lines indicate the continuous transition lines.

Figure 2

Each row contains a surface and a contour plot of the exact probability distribution for the values of $\alpha$ and $\beta$ indicated, and with $\rho =N/L$ and $j=J/L$. The first three rows illustrate the behavior of the probability distribution as the system goes along the line of $\alpha =\beta$ through the critical point at $\alpha =\beta =0.5$, while the last three graphs illustrate the behavior as the system goes through the first order transition at $\alpha =\beta =0.25$. Overlaid in the contour plots (dashed line) is the curve $j=\rho (1-\rho )$ which defines the set of possible asymptotic average values of $\rho$ and $j$ throughout the system's different phases (not at the first order transition line). The system size is $L=40$.

Figure 3

The two rows display a surface and a contour plot of the leading behavior of the asymptotic probability distribution. The calculations were performed at the injection and ejection rates indicated and at a system of size $L=40$ (to make the result comparable to Fig. 2).